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lecture09 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 1 ApplicationsofKKTConditionsandNonlinearOptimization YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/˜yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 2 PortfolioManagement Let r denotethe expectedreturnvector and V denotethe co-variancematrix of aninvestmentportfolio,andlet x betheinvestmentproportionvector. minimize x T V x subjectto r T x ≥ π, e T x =1 , x ≥ , where e isthevectorofallones.Thisisa quadraticprogram . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 3 AllowShortOption minimize x T V x subjectto r T x ≥ π, e T x =1 . OptimalityConditions: 2 V x- λ r- μ e = ,λ ≥ ,λ ( r T x- π )=0 . μ =2 x T V x- λ · π. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 4 Solutiontolinearequations Trycase r T x >π ,thatis, λ =0 : 2 V- e e T x μ = 1 andverifywhetherornot r T x >π ;ifnotsolve 2 V- r- e r T e T x λ μ = π 1 . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 5 PortfolioExample r 1 =0 . 0825 ,r 2 =0 . 1125 , V = . 01390 . 0056 . 00560 . 0261 π =0 . 1 . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 6 KnapsackMaximization maximize ∑ j f j ( x j ) subjectto ∑ j p j x j ≤ w, x j ≥ , ∀ j OptimalityConditions: λ- f j ( x j ) p j ≥ ,λ ≥ ,x j λ- f j ( x j ) p j ¶ =0 ,λ ( w- X j p j x j )=0 , ∀ j. Implications: λ = f j ( x j ) p j , ∀ x j > 0; and λ ≥ f j ( x j ) p j , ∀ x j =0 Investonlytotheprojectswhose marginalreturnperdollar isaboveathreshold λ . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 7 Fisher’sExchangeMarket Buyers havemoney( w i...
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This note was uploaded on 05/06/2010 for the course MSE 211 at Stanford.

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lecture09 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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